Finding the smallest σ algebra containing two subsets

[gimmytang]

Hi guys,
I am new comer here. I am now stuyding mathematical analysis and have a bunch of questions. Is here the right place for seeking help from people?
Ok.. Here comes my real question in $$\sigma$$ algebra:
Let $$A_1,A_2$$ be two arbitrary subsets of $$\Omega$$, find the smallest $$\sigma$$ algebra containing $$A_1,A_2$$.

Thanks!
gimmy

 

[matt grime]

A sigma algebra is closed under what operations?

 

[gimmytang]

It is closed under complementation and finite union. Therefore the smallest $$\sigma$$ algebra $$\chi =\{\Phi,\Omega, A_{1}, A_{1}^{c},A_{2},A_{2}^{c}\}$$. Is it correct?

 

[matt grime]

"closed under finite union" so where is (A_1)u(A_2)?

 

[gimmytang]

yeah, should be more sets inside like this:
$$\{\Phi,\Omega,A_1,A_{1}^{c},A_2,A_{2}^{c},A_{1}\cup A_{2},A_{1}\cup A_{2}^{c},A_{1}^{c}\cup A_{2},A_{1}^{c}\cup A_{2}^{c},A_{1}\cap A_{2},A_{1}\cap A_{2}^{c},A_{1}^{c}\cap A_{2},A_{1}^{c}\cap A_{2}^{c}\}$$

 

[Hurkyl]

Aren't σ-algebras closed under countable unions? (It doesn't matter for this example, though)

 

[master_coda]

Aren't σ-algebras closed under countable unions? (It doesn't matter for this example, though)

Yes, they are.

 

[gimmytang]

But according to the answer in a book the number of members in the smallest $$\sigma$$-field containing $$A_1,...A_n$$ is $$2^{2^n}$$. For n=2, there will be 16 members, while I only get 14. What the other 2 memebers are supposed to be?

 

[matt grime]

How about {(A_1)n(A_2)}u{(A_1^c)n(A_2^c)}?

Although it pains me to say it, look at a Venn diagram.

 

[gimmytang]

I got it. It is easy to see after drawing a Venn diagram. There are four partitions on the diagram. So the possible sigma field is any combination of the four partitions, which has the number of 16 in total.
Thank you all!